We study the restrictions, the strict fixed points, and the strict quotients of the partition complex \(|\Pi_{n}|\), which is the \(\Sigma_{n}\)-space attached to the poset of proper nontrivial partitions of the set \(\{1,\ldots,n\}\).

We express the space of fixed points \(|\Pi_{n}|^{G}\) in terms of subgroup posets for general \(G\subset \Sigma_{n}\) and prove a formula for the restriction of \(|\Pi_{n}|\) to Young subgroups \(\Sigma_{n_{1}}\times \cdots\times \Sigma_{n_{k}}\). Both results follow by applying a general method, proven with discrete Morse theory, for producing equivariant branching rules on lattices with group actions.

We uncover surprising links between strict Young quotients of \(|\Pi_{n}|\), commutative monoid spaces, and the cotangent fibre in derived algebraic geometry. These connections allow us to construct a cofibre sequence relating various strict quotients \(|\Pi_{n}|^{\diamond} \mathbin {\operatorname* {\wedge }_{\Sigma_{n}}^{}} (S^{\ell})^{\wedge n}\) and give a combinatorial proof of a splitting in derived algebraic geometry.

Combining all our results, we decompose strict Young quotients of \(|\Pi_{n}|\) in terms of “atoms” \(|\Pi_{d}|^{\diamond} \mathbin {\operatorname* {\wedge }_{\Sigma_{d}}^{}} (S^{\ell})^{\wedge d}\) for \(\ell\) odd and compute their homology. We thereby also generalise Goerss’ computation of the algebraic André-Quillen homology of trivial square-zero extensions from \(\mathbf {F}_{2}\) to \(\mathbf {F}_{p}\) for \(p\) an odd prime.

我们研究分区复合物\（| \ Pi_ {n} | \）的限制，严格的不动点和严格的商，\（| \ Pi_ {n} | \）是附加到适当的位姿的\（\ Sigma_ {n} \） -空间集合\（\ {1，\ ldots，n \} \）的非平凡分区。

我们用一般\（G \ subset \ Sigma_ {n} \）的子组姿态表示固定点\（|| Pi_ {n} | ^ {G} \）的空间，并证明限制\的公式（| \ Pi_ {n} | \）到Young子组\（\ Sigma_ {n_ {1}} \ times \ cdots \ times \ Sigma_ {n_ {k}} \\）。两种结果都采用了一种通用的方法，该方法已通过离散莫尔斯理论进行了证明，可以在具有群作用的晶格上产生等变分支规则。

我们发现\（| \ Pi_ {n} | \）的严格Young商，可交换单调空间和衍生代数几何中的余切纤维之间的令人惊讶的联系。这些连接使我们能够构造与各种严格商\（| \ Pi_ {n} | ^ {\ diamond} \ mathbin {\ operatorname * {\ wedge} _ {\ Sigma_ {n}} ^ {}}（ S ^ {\ ell}）^ {\ wedge n} \）并给出派生代数几何分裂的组合证明。

结合所有结果，我们分解严格的年轻商\（| \ Pi_ {N} | \）在“原子”的条款\（| \ Pi_ {d} | ^ {\钻石} \ mathbin {\ operatorname * {\楔} _ {\ Sigma_ {d}} ^ {}}（S ^ {\ ELL}）^ {\楔d} \）为\（\ ELL \）奇数和计算它们的同源性。我们由此也概括琐碎方零扩展的代数安德烈-奎伦同源性Goerss'运算从\（\ mathbf {F} _ {2} \）到\（\ mathbf {F} _ {P} \）为\ （p \）一个奇数素数。